The survey by Lord Ashcroft, the former Conservative deputy chairman, gives Labour a 14-point lead in the 32 seats which have the smallest current Tory majorities over Mr Miliband’s party.

Labour’s 14-point advantage in their target seats compares with a national lead of five points (35 per cent to 30 per cent) over the Tories in a “comparison” poll covering the whole country also undertaken by the peer.

The huge marginals poll on the eve of the party conference season, which covers nearly 13,000 voters in 40 Conservative-held constituencies, does offer some comfort to David Cameron, who is seen as a better choice as prime minister by 13 points over Mr Miliband.

It also suggests that at the next election, in May 2015, the Tories will hold seats where they are currently defending narrow majorities from Liberal Democrats in second place.

However, it is the surge by Ukip, taking votes from the Conservative and letting in Labour, which is causing the most disquiet at 10 Downing Street and Conservative Campaign Headquarters. No political party currently has the money to compile regular polls in key marginals on the scale of today’s survey by Lord Ashcroft, the first of its kind for two years.

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The peer said Mr Miliband could become prime minister with a “comfortable majority” if the Ukip surge held. The Tories should concentrate on showing they were “on the side of ordinary voters,” he added.

Grant Shapps, the Conservative Party Chairman, has unveiled an “40/40” election strategy which sees resources concentrated on the 40 most vulnerable Tory-held seats and 40 other constituencies in which they are close challengers to the sitting MP.

Lord Ashcroft’s poll covers the 40 Tory seats with the slimmest majorities - and puts UKIP support in them in double figures, more than tripling the average three per cent vote share recorded by Nigel Farage’s party in the 2010 election.

Ukip’s surge has helped squeeze the Tory vote and provide an 8.5 per cent swing to Labour since the 2010 general election in the 32 key marginals, enough for Mr Miliband’s party to win all of them.

In the 32 seats Labour led the Conservatives by 43 per cent to 29 per cent - a result likely to hand Labour a Commons majority of around 60 per cent.

Ukip were on 11 per cent with support for the Lib Dems slumping to just eight per cent.

In eight Tory-held seats which are Lib-Dem targets, the poll results suggest Mr Cameron’s party will hold all of them, despite Ukip’s strong showing. In these seats the Conservatives are on 32 per cent, with the Lib Dems on 29 per cent, Labour on 18 and Ukip on 12.

Asked who would make the best prime minister, 39 per cent of voters across all 40 marginals chose Mr Cameron, with 26 per cent opting for Mr Milibandand 12 per cent for Nick Clegg.

Labour will take some cheer from the poll which comes after a bruising summer for Mr Miliband in which his leadership came under attack from his own side.

However, the survey shows Labour has made no further progress in its Tory-held targets since 2011. In a similar poll two years ago, Lord Ashcroft found Labour on 44 per cent in their targets, compared to 43 per cent today. Support for the Tories over the same period has slumped by six points from 35 per cent to 29 per cent.

Labour was in the lead on the economy and jobs, the NHS and education while the Lib Dems were rated best on only one issue - the environment.

Marginal voters were more likely to see the Tories as “clear about where they stand for,” “willing to take tough decisions for the long term” and “competent and capable.” Labour were most likely to “share my values” and “be concerned about people like me”.

On how to reduce the deficit, 45 per cent agreed with Labour’s broad view - that government spending cuts were “too deep and too quick.” However, 40 per cent believed the reductions were “necessary, avoidable and right.” one in 10 voters though the cutbacks did not go far enough.

Overall, just over half (52 per cent) thought Britain was going “in the right direction” with 45 per cent thinking the country was heading in the wrong direction.

Some 42 per cent thought the economy would “improve significantly in the next two years because the “right decisions” were being made now but 56 per cent thought things would either show no improvement or get worse by 2015.

Lord Ashcroft said: “Though this is only a snapshot, the picture is clear - despite their narrow national poll lead, Labour are further ahead in the marginal where it matters. Labour have made progress in the last two years, but if Ukip does as well at the general election as this poll suggests Ed Miliband could become prime ministers with comfortable majority.

“But it is not close to being over yet. Labour have to establish real trust on managing the economy, David Cameron is still seen as the best prime minister and more people say Britain is heading in the right direction than say the opposite.

“The debate will continue within the party as to how far the Tories can tempt voters back by being more like Ukip, and how many will be persuaded that by voting for a fringe party they will open the door to Prime Minister Miliband. But the priority must be to be a competent government with the right priorities and show that the Conservatives are on the side of ordinary voters.

“With 20 months to go until polling day I think we are in a for a close and exciting election.”

Polls of marginal constituencies have played a key role in framing decision-making in the past, most notably when a survey of marginals in the autumn of 2007 helped persuade Gordon Brown not to stage a snap general election.

*12,809 adults were interviewed by telephone in the 40 most marginal Conservative-held seats between 1 August and 5 September 2013. 300 interviews were conducted in each of the 32 seats where Labour were second at the 2010 general election, and 400 in each of the 8 seats where the Lib Dems were second.

For comparison, 1,005 adults throughout Britain were interviewed by telephone between 30 August and 1 September 2013.